Question: $\dfrac{ 3u + 5v }{ 6 } = \dfrac{ u - 5w }{ -8 }$ Solve for $u$.
Multiply both sides by the left denominator. $\dfrac{ 3u + 5v }{ {6} } = \dfrac{ u - 5w }{ -8 }$ ${6} \cdot \dfrac{ 3u + 5v }{ {6} } = {6} \cdot \dfrac{ u - 5w }{ -8 }$ $3u + 5v = {6} \cdot \dfrac { u - 5w }{ -8 }$ Multiply both sides by the right denominator. $3u + 5v = 6 \cdot \dfrac{ u - 5w }{ -{8} }$ $-{8} \cdot \left( 3u + 5v \right) = -{8} \cdot 6 \cdot \dfrac{ u - 5w }{ -{8} }$ $-{8} \cdot \left( 3u + 5v \right) = 6 \cdot \left( u - 5w \right)$ Distribute both sides $-{8} \cdot \left( 3u + 5v \right) = {6} \cdot \left( u - 5w \right)$ $-{24}u - {40}v = {6}u - {30}w$ Combine $u$ terms on the left. $-{24u} - 40v = {6u} - 30w$ $-{30u} - 40v = -30w$ Move the $v$ term to the right. $-30u - {40v} = -30w$ $-30u = -30w + {40v}$ Isolate $u$ by dividing both sides by its coefficient. $-{30}u = -30w + 40v$ $u = \dfrac{ -30w + 40v }{ -{30} }$ All of these terms are divisible by $10$ Divide by the common factor and swap signs so the denominator isn't negative. $u = \dfrac{ {3}w - {4}v }{ {3} }$